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Chapter 4: Problem 9

Divide. $$ \frac{15 m^{3}+25 m^{2}+30 m}{5 m^{3}} $$

Short Answer

Expert verified
3 + \frac{5}{m} + \frac{6}{m^2}

Step by step solution

01

Simplify the Fraction

First, rewrite the given expression as a single fraction: \[ \frac{15m^3 + 25m^2 + 30m}{5m^3} \]
02

Divide Each Term Individually

Next, divide each term in the numerator by the term in the denominator individually: \[ \frac{15m^3}{5m^3} + \frac{25m^2}{5m^3} + \frac{30m}{5m^3} \]
03

Simplify Each Fraction

Now, simplify each fractional expression: \[ \frac{15m^3}{5m^3} = 3 \] \[ \frac{25m^2}{5m^3} = \frac{25}{5} \frac{m^2}{m^3} = 5 \frac{1}{m} = \frac{5}{m} \] \[ \frac{30m}{5m^3} = \frac{30}{5} \frac{m}{m^3} = 6 \frac{1}{m^2} = \frac{6}{m^2} \]
04

Combine Simplified Terms

Finally, combine the simplified terms: \[ 3 + \frac{5}{m} + \frac{6}{m^2} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Fraction Simplification
To solve the given problem, understanding fraction simplification is essential. In math, simplification means breaking down fractions to their most basic form. Each term within the fraction has common factors that can be reduced. Here, we had the fraction: \[ \frac{15m^3 + 25m^2 + 30m}{5m^3} \] We treated it as a sum of individual fractions: \[ \frac{15m^3}{5m^3} + \frac{25m^2}{5m^3} + \frac{30m}{5m^3} \] By simplifying each of these terms, we ended up with simpler fractions: \[ 3 + \frac{5}{m} + \frac{6}{m^2} \] Simplifying each of these means making them easier to work with by breaking down the numbers and variables to the smallest possible form.
Polynomial Division
Polynomial division involves dividing one polynomial by another. When you have a polynomial divided by a monomial (single term), it鈥檚 straightforward once you break it down. In the given problem: \[ \frac{15m^3 + 25m^2 + 30m}{5m^3} \] We divided each term individually by the denominator:
  • \[ \frac{15m^3}{5m^3} = 3 \]
  • \[ \frac{25m^2}{5m^3} = \frac{5}{m} \]
  • \[ \frac{30m}{5m^3} = \frac{6}{m^2} \]
Breaking it down into individual terms makes the division easier. This process is known as term-by-term division and is crucial for understanding polynomial division in algebra.
Rational Functions
Rational functions are expressions that involve ratios of polynomials. Understanding these types of functions is key to tackling exercises involving rational expressions. In our example, the rational function was given by: \[ \frac{15m^3 + 25m^2 + 30m}{5m^3} \] We transformed this into simplified terms: \[ 3 + \frac{5}{m} + \frac{6}{m^2} \] This shows how rational functions can often be simplified into more manageable pieces. Each term is a fraction involving a polynomial's single term. Recognizing and simplifying rational functions is pivotal in algebra, making complex problems easier to solve.

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Most popular questions from this chapter

Simplify each expression. Assume that all variables represent nonzero real numbers. $$ \left(\frac{5 m^{4} n^{-3}}{m^{-5} n^{2}}\right)^{-2} $$

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Find each product. $$ (5 r-s)^{3} $$
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